LMFDB - Embedded newform 1392.2.o.i.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1392,2,Mod(289,1392)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1392, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1392.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1392 = 2^{4} \cdot 3 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1392.o (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.1151759614$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 87)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1392.289
Dual form			1392.2.o.i.289.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.23607 q^{5} +4.23607 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.23607 q^{5} +4.23607 q^{7} -1.00000 q^{9} -2.23607i q^{11} -1.00000 q^{13} -3.23607i q^{15} -3.00000i q^{17} -7.70820i q^{19} -4.23607i q^{21} -1.23607 q^{23} +5.47214 q^{25} +1.00000i q^{27} +(-4.47214 + 3.00000i) q^{29} +7.23607i q^{31} -2.23607 q^{33} +13.7082 q^{35} +4.76393i q^{37} +1.00000i q^{39} -4.47214i q^{41} +0.472136i q^{43} -3.23607 q^{45} +10.2361i q^{47} +10.9443 q^{49} -3.00000 q^{51} +6.76393 q^{53} -7.23607i q^{55} -7.70820 q^{57} -8.94427 q^{59} +2.76393i q^{61} -4.23607 q^{63} -3.23607 q^{65} -4.70820 q^{67} +1.23607i q^{69} -4.76393 q^{71} -7.70820i q^{73} -5.47214i q^{75} -9.47214i q^{77} +2.29180i q^{79} +1.00000 q^{81} +6.00000 q^{83} -9.70820i q^{85} +(3.00000 + 4.47214i) q^{87} -0.0557281i q^{89} -4.23607 q^{91} +7.23607 q^{93} -24.9443i q^{95} +18.1803i q^{97} +2.23607i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 8 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 8 * q^7 - 4 * q^9	$ 4 q + 4 q^{5} + 8 q^{7} - 4 q^{9} - 4 q^{13} + 4 q^{23} + 4 q^{25} + 28 q^{35} - 4 q^{45} + 8 q^{49} - 12 q^{51} + 36 q^{53} - 4 q^{57} - 8 q^{63} - 4 q^{65} + 8 q^{67} - 28 q^{71} + 4 q^{81} + 24 q^{83} + 12 q^{87} - 8 q^{91} + 20 q^{93}+O(q^{100}) $ 4 * q + 4 * q^5 + 8 * q^7 - 4 * q^9 - 4 * q^13 + 4 * q^23 + 4 * q^25 + 28 * q^35 - 4 * q^45 + 8 * q^49 - 12 * q^51 + 36 * q^53 - 4 * q^57 - 8 * q^63 - 4 * q^65 + 8 * q^67 - 28 * q^71 + 4 * q^81 + 24 * q^83 + 12 * q^87 - 8 * q^91 + 20 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1392\mathbb{Z}\right)^\times$.

$n$	$175$	$929$	$1045$	$1249$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	3.23607			1.44721			0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607			1.44721			0.723607	+	0.690212i	$0.242483\pi$
$6$		0			0
$7$	4.23607			1.60108			0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607			1.60108			0.800542	+	0.599277i	$0.204545\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	2.23607i		−	0.674200i	−0.941469	−	0.337100i	$-0.890554\pi$
$11$		−	2.23607i		−	0.674200i	0.941469	−	0.337100i	$-0.109446\pi$
$12$		0			0
$13$	−1.00000			−0.277350			−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000			−0.277350			−0.138675	+	0.990338i	$0.544284\pi$
$14$		0			0
$15$		−	3.23607i		−	0.835549i
$16$		0			0
$17$		−	3.00000i		−	0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$		−	3.00000i		−	0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$		0			0
$19$		−	7.70820i		−	1.76838i	−0.467124	−	0.884192i	$-0.654710\pi$
$19$		−	7.70820i		−	1.76838i	0.467124	−	0.884192i	$-0.345290\pi$
$20$		0			0
$21$		−	4.23607i		−	0.924386i
$22$		0			0
$23$	−1.23607			−0.257738			−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607			−0.257738			−0.128869	+	0.991662i	$0.541135\pi$
$24$		0			0
$25$	5.47214			1.09443
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−4.47214	+	3.00000i	−0.830455	+	0.557086i
$29$	−4.47214	+	3.00000i	−0.830455	+	0.557086i
$30$		0			0
$31$			7.23607i			1.29964i	0.760090	+	0.649818i	$0.225155\pi$
$31$			7.23607i			1.29964i	−0.760090	+	0.649818i	$0.774845\pi$
$32$		0			0
$33$	−2.23607			−0.389249
$34$		0			0
$35$	13.7082			2.31711
$36$		0			0
$37$			4.76393i			0.783186i	0.920139	+	0.391593i	$0.128076\pi$
$37$			4.76393i			0.783186i	−0.920139	+	0.391593i	$0.871924\pi$
$38$		0			0
$39$			1.00000i			0.160128i
$40$		0			0
$41$		−	4.47214i		−	0.698430i	−0.937043	−	0.349215i	$-0.886448\pi$
$41$		−	4.47214i		−	0.698430i	0.937043	−	0.349215i	$-0.113552\pi$
$42$		0			0
$43$			0.472136i			0.0720001i	0.999352	+	0.0360000i	$0.0114616\pi$
$43$			0.472136i			0.0720001i	−0.999352	+	0.0360000i	$0.988538\pi$
$44$		0			0
$45$	−3.23607			−0.482405
$46$		0			0
$47$			10.2361i			1.49308i	0.665338	+	0.746542i	$0.268287\pi$
$47$			10.2361i			1.49308i	−0.665338	+	0.746542i	$0.731713\pi$
$48$		0			0
$49$	10.9443			1.56347
$50$		0			0
$51$	−3.00000			−0.420084
$52$		0			0
$53$	6.76393			0.929098			0.464549	−	0.885548i	$-0.346217\pi$
$53$	6.76393			0.929098			0.464549	+	0.885548i	$0.346217\pi$
$54$		0			0
$55$		−	7.23607i		−	0.975711i
$56$		0			0
$57$	−7.70820			−1.02098
$58$		0			0
$59$	−8.94427			−1.16445			−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427			−1.16445			−0.582223	+	0.813029i	$0.697817\pi$
$60$		0			0
$61$			2.76393i			0.353885i	0.984221	+	0.176943i	$0.0566207\pi$
$61$			2.76393i			0.353885i	−0.984221	+	0.176943i	$0.943379\pi$
$62$		0			0
$63$	−4.23607			−0.533694
$64$		0			0
$65$	−3.23607			−0.401385
$66$		0			0
$67$	−4.70820			−0.575199			−0.287599	−	0.957751i	$-0.592857\pi$
$67$	−4.70820			−0.575199			−0.287599	+	0.957751i	$0.592857\pi$
$68$		0			0
$69$			1.23607i			0.148805i
$70$		0			0
$71$	−4.76393			−0.565375			−0.282687	−	0.959212i	$-0.591226\pi$
$71$	−4.76393			−0.565375			−0.282687	+	0.959212i	$0.591226\pi$
$72$		0			0
$73$		−	7.70820i		−	0.902177i	−0.892479	−	0.451089i	$-0.851036\pi$
$73$		−	7.70820i		−	0.902177i	0.892479	−	0.451089i	$-0.148964\pi$
$74$		0			0
$75$		−	5.47214i		−	0.631868i
$76$		0			0
$77$		−	9.47214i		−	1.07945i
$78$		0			0
$79$			2.29180i			0.257847i	0.991655	+	0.128924i	$0.0411522\pi$
$79$			2.29180i			0.257847i	−0.991655	+	0.128924i	$0.958848\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$		−	9.70820i		−	1.05300i
$86$		0			0
$87$	3.00000	+	4.47214i	0.321634	+	0.479463i
$88$		0			0
$89$		−	0.0557281i		−	0.00590717i	−0.999996	−	0.00295358i	$-0.999060\pi$
$89$		−	0.0557281i		−	0.00590717i	0.999996	−	0.00295358i	$-0.000940156\pi$
$90$		0			0
$91$	−4.23607			−0.444061
$92$		0			0
$93$	7.23607			0.750345
$94$		0			0
$95$		−	24.9443i		−	2.55923i
$96$		0			0
$97$			18.1803i			1.84593i	0.384879	+	0.922967i	$0.374243\pi$
$97$			18.1803i			1.84593i	−0.384879	+	0.922967i	$0.625757\pi$
$98$		0			0
$99$			2.23607i			0.224733i
$100$		0			0
$101$		−	3.94427i		−	0.392470i	−0.980557	−	0.196235i	$-0.937129\pi$
$101$		−	3.94427i		−	0.392470i	0.980557	−	0.196235i	$-0.0628715\pi$
$102$		0			0
$103$	19.4164			1.91316			0.956578	−	0.291477i	$-0.0941468\pi$
$103$	19.4164			1.91316			0.956578	+	0.291477i	$0.0941468\pi$
$104$		0			0
$105$		−	13.7082i		−	1.33778i
$106$		0			0
$107$	−0.763932			−0.0738521			−0.0369260	−	0.999318i	$-0.511757\pi$
$107$	−0.763932			−0.0738521			−0.0369260	+	0.999318i	$0.511757\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$		0			0
$111$	4.76393			0.452172
$112$		0			0
$113$			4.52786i			0.425946i	0.977058	+	0.212973i	$0.0683146\pi$
$113$			4.52786i			0.425946i	−0.977058	+	0.212973i	$0.931685\pi$
$114$		0			0
$115$	−4.00000			−0.373002
$116$		0			0
$117$	1.00000			0.0924500
$118$		0			0
$119$		−	12.7082i		−	1.16496i
$120$		0			0
$121$	6.00000			0.545455
$122$		0			0
$123$	−4.47214			−0.403239
$124$		0			0
$125$	1.52786			0.136656
$126$		0			0
$127$			12.4721i			1.10672i	0.832941	+	0.553362i	$0.186655\pi$
$127$			12.4721i			1.10672i	−0.832941	+	0.553362i	$0.813345\pi$
$128$		0			0
$129$	0.472136			0.0415693
$130$		0			0
$131$		−	12.2361i		−	1.06907i	−0.845146	−	0.534535i	$-0.820487\pi$
$131$		−	12.2361i		−	1.06907i	0.845146	−	0.534535i	$-0.179513\pi$
$132$		0			0
$133$		−	32.6525i		−	2.83133i
$134$		0			0
$135$			3.23607i			0.278516i
$136$		0			0
$137$		−	3.52786i		−	0.301406i	−0.988579	−	0.150703i	$-0.951846\pi$
$137$		−	3.52786i		−	0.301406i	0.988579	−	0.150703i	$-0.0481537\pi$
$138$		0			0
$139$	6.70820			0.568982			0.284491	−	0.958679i	$-0.408175\pi$
$139$	6.70820			0.568982			0.284491	+	0.958679i	$0.408175\pi$
$140$		0			0
$141$	10.2361			0.862032
$142$		0			0
$143$			2.23607i			0.186989i
$144$		0			0
$145$	−14.4721	+	9.70820i	−1.20185	+	0.806222i
$146$		0			0
$147$		−	10.9443i		−	0.902668i
$148$		0			0
$149$	−5.52786			−0.452860			−0.226430	−	0.974027i	$-0.572705\pi$
$149$	−5.52786			−0.452860			−0.226430	+	0.974027i	$0.572705\pi$
$150$		0			0
$151$	−12.0000			−0.976546			−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000			−0.976546			−0.488273	+	0.872691i	$0.662373\pi$
$152$		0			0
$153$			3.00000i			0.242536i
$154$		0			0
$155$			23.4164i			1.88085i
$156$		0			0
$157$			2.00000i			0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$			2.00000i			0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$		0			0
$159$		−	6.76393i		−	0.536415i
$160$		0			0
$161$	−5.23607			−0.412660
$162$		0			0
$163$		−	7.41641i		−	0.580898i	−0.956890	−	0.290449i	$-0.906195\pi$
$163$		−	7.41641i		−	0.580898i	0.956890	−	0.290449i	$-0.0938046\pi$
$164$		0			0
$165$	−7.23607			−0.563327
$166$		0			0
$167$	−15.8885			−1.22949			−0.614746	−	0.788725i	$-0.710742\pi$
$167$	−15.8885			−1.22949			−0.614746	+	0.788725i	$0.710742\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$			7.70820i			0.589461i
$172$		0			0
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$	23.1803			1.75227
$176$		0			0
$177$			8.94427i			0.672293i
$178$		0			0
$179$	17.2361			1.28828			0.644142	−	0.764906i	$-0.277215\pi$
$179$	17.2361			1.28828			0.644142	+	0.764906i	$0.277215\pi$
$180$		0			0
$181$	20.4164			1.51754			0.758770	−	0.651359i	$-0.225801\pi$
$181$	20.4164			1.51754			0.758770	+	0.651359i	$0.225801\pi$
$182$		0			0
$183$	2.76393			0.204316
$184$		0			0
$185$			15.4164i			1.13344i
$186$		0			0
$187$	−6.70820			−0.490552
$188$		0			0
$189$			4.23607i			0.308129i
$190$		0			0
$191$		−	8.94427i		−	0.647185i	−0.946197	−	0.323592i	$-0.895109\pi$
$191$		−	8.94427i		−	0.647185i	0.946197	−	0.323592i	$-0.104891\pi$
$192$		0			0
$193$			17.4164i			1.25366i	0.779156	+	0.626830i	$0.215648\pi$
$193$			17.4164i			1.25366i	−0.779156	+	0.626830i	$0.784352\pi$
$194$		0			0
$195$			3.23607i			0.231740i
$196$		0			0
$197$	−3.70820			−0.264199			−0.132099	−	0.991236i	$-0.542172\pi$
$197$	−3.70820			−0.264199			−0.132099	+	0.991236i	$0.542172\pi$
$198$		0			0
$199$	20.1246			1.42660			0.713298	−	0.700861i	$-0.247201\pi$
$199$	20.1246			1.42660			0.713298	+	0.700861i	$0.247201\pi$
$200$		0			0
$201$			4.70820i			0.332091i
$202$		0			0
$203$	−18.9443	+	12.7082i	−1.32963	+	0.891941i
$204$		0			0
$205$		−	14.4721i		−	1.01078i
$206$		0			0
$207$	1.23607			0.0859127
$208$		0			0
$209$	−17.2361			−1.19224
$210$		0			0
$211$		−	10.0000i		−	0.688428i	−0.938891	−	0.344214i	$-0.888145\pi$
$211$		−	10.0000i		−	0.688428i	0.938891	−	0.344214i	$-0.111855\pi$
$212$		0			0
$213$			4.76393i			0.326419i
$214$		0			0
$215$			1.52786i			0.104199i
$216$		0			0
$217$			30.6525i			2.08083i
$218$		0			0
$219$	−7.70820			−0.520872
$220$		0			0
$221$			3.00000i			0.201802i
$222$		0			0
$223$	12.7082			0.851004			0.425502	−	0.904957i	$-0.360097\pi$
$223$	12.7082			0.851004			0.425502	+	0.904957i	$0.360097\pi$
$224$		0			0
$225$	−5.47214			−0.364809
$226$		0			0
$227$	−9.05573			−0.601050			−0.300525	−	0.953774i	$-0.597162\pi$
$227$	−9.05573			−0.601050			−0.300525	+	0.953774i	$0.597162\pi$
$228$		0			0
$229$			22.1803i			1.46572i	0.680380	+	0.732859i	$0.261815\pi$
$229$			22.1803i			1.46572i	−0.680380	+	0.732859i	$0.738185\pi$
$230$		0			0
$231$	−9.47214			−0.623221
$232$		0			0
$233$	−10.4721			−0.686052			−0.343026	−	0.939326i	$-0.611452\pi$
$233$	−10.4721			−0.686052			−0.343026	+	0.939326i	$0.611452\pi$
$234$		0			0
$235$			33.1246i			2.16081i
$236$		0			0
$237$	2.29180			0.148868
$238$		0			0
$239$	−26.1803			−1.69347			−0.846733	−	0.532019i	$-0.821434\pi$
$239$	−26.1803			−1.69347			−0.846733	+	0.532019i	$0.821434\pi$
$240$		0			0
$241$	−3.00000			−0.193247			−0.0966235	−	0.995321i	$-0.530804\pi$
$241$	−3.00000			−0.193247			−0.0966235	+	0.995321i	$0.530804\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	35.4164			2.26267
$246$		0			0
$247$			7.70820i			0.490461i
$248$		0			0
$249$		−	6.00000i		−	0.380235i
$250$		0			0
$251$			12.2361i			0.772334i	0.922429	+	0.386167i	$0.126201\pi$
$251$			12.2361i			0.772334i	−0.922429	+	0.386167i	$0.873799\pi$
$252$		0			0
$253$			2.76393i			0.173767i
$254$		0			0
$255$	−9.70820			−0.607951
$256$		0			0
$257$	−26.4721			−1.65129			−0.825643	−	0.564193i	$-0.809188\pi$
$257$	−26.4721			−1.65129			−0.825643	+	0.564193i	$0.809188\pi$
$258$		0			0
$259$			20.1803i			1.25395i
$260$		0			0
$261$	4.47214	−	3.00000i	0.276818	−	0.185695i
$262$		0			0
$263$			4.94427i			0.304877i	0.988313	+	0.152438i	$0.0487126\pi$
$263$			4.94427i			0.304877i	−0.988313	+	0.152438i	$0.951287\pi$
$264$		0			0
$265$	21.8885			1.34460
$266$		0			0
$267$	−0.0557281			−0.00341050
$268$		0			0
$269$		−	21.3607i		−	1.30238i	−0.758913	−	0.651192i	$-0.774269\pi$
$269$		−	21.3607i		−	1.30238i	0.758913	−	0.651192i	$-0.225731\pi$
$270$		0			0
$271$			23.4164i			1.42245i	0.702967	+	0.711223i	$0.251858\pi$
$271$			23.4164i			1.42245i	−0.702967	+	0.711223i	$0.748142\pi$
$272$		0			0
$273$			4.23607i			0.256378i
$274$		0			0
$275$		−	12.2361i		−	0.737863i
$276$		0			0
$277$	6.41641			0.385525			0.192762	−	0.981245i	$-0.438255\pi$
$277$	6.41641			0.385525			0.192762	+	0.981245i	$0.438255\pi$
$278$		0			0
$279$		−	7.23607i		−	0.433212i
$280$		0			0
$281$	−1.41641			−0.0844958			−0.0422479	−	0.999107i	$-0.513452\pi$
$281$	−1.41641			−0.0844958			−0.0422479	+	0.999107i	$0.513452\pi$
$282$		0			0
$283$	13.8885			0.825588			0.412794	−	0.910824i	$-0.364553\pi$
$283$	13.8885			0.825588			0.412794	+	0.910824i	$0.364553\pi$
$284$		0			0
$285$	−24.9443			−1.47757
$286$		0			0
$287$		−	18.9443i		−	1.11825i
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$	18.1803			1.06575
$292$		0			0
$293$		−	29.9443i		−	1.74936i	−0.484698	−	0.874682i	$-0.661071\pi$
$293$		−	29.9443i		−	1.74936i	0.484698	−	0.874682i	$-0.338929\pi$
$294$		0			0
$295$	−28.9443			−1.68520
$296$		0			0
$297$	2.23607			0.129750
$298$		0			0
$299$	1.23607			0.0714837
$300$		0			0
$301$			2.00000i			0.115278i
$302$		0			0
$303$	−3.94427			−0.226593
$304$		0			0
$305$			8.94427i			0.512148i
$306$		0			0
$307$		−	28.1803i		−	1.60834i	−0.594401	−	0.804168i	$-0.702611\pi$
$307$		−	28.1803i		−	1.60834i	0.594401	−	0.804168i	$-0.297389\pi$
$308$		0			0
$309$		−	19.4164i		−	1.10456i
$310$		0			0
$311$			15.6525i			0.887570i	0.896133	+	0.443785i	$0.146365\pi$
$311$			15.6525i			0.887570i	−0.896133	+	0.443785i	$0.853635\pi$
$312$		0			0
$313$	−5.47214			−0.309303			−0.154652	−	0.987969i	$-0.549426\pi$
$313$	−5.47214			−0.309303			−0.154652	+	0.987969i	$0.549426\pi$
$314$		0			0
$315$	−13.7082			−0.772370
$316$		0			0
$317$			23.8328i			1.33858i	0.742999	+	0.669292i	$0.233403\pi$
$317$			23.8328i			1.33858i	−0.742999	+	0.669292i	$0.766597\pi$
$318$		0			0
$319$	6.70820	+	10.0000i	0.375587	+	0.559893i
$320$		0			0
$321$			0.763932i			0.0426385i
$322$		0			0
$323$	−23.1246			−1.28669
$324$		0			0
$325$	−5.47214			−0.303539
$326$		0			0
$327$		−	5.00000i		−	0.276501i
$328$		0			0
$329$			43.3607i			2.39055i
$330$		0			0
$331$			8.29180i			0.455758i	0.973689	+	0.227879i	$0.0731791\pi$
$331$			8.29180i			0.455758i	−0.973689	+	0.227879i	$0.926821\pi$
$332$		0			0
$333$		−	4.76393i		−	0.261062i
$334$		0			0
$335$	−15.2361			−0.832435
$336$		0			0
$337$			19.2361i			1.04786i	0.851763	+	0.523928i	$0.175534\pi$
$337$			19.2361i			1.04786i	−0.851763	+	0.523928i	$0.824466\pi$
$338$		0			0
$339$	4.52786			0.245920
$340$		0			0
$341$	16.1803			0.876215
$342$		0			0
$343$	16.7082			0.902158
$344$		0			0
$345$			4.00000i			0.215353i
$346$		0			0
$347$	22.6525			1.21605			0.608024	−	0.793918i	$-0.291962\pi$
$347$	22.6525			1.21605			0.608024	+	0.793918i	$0.291962\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$		−	1.00000i		−	0.0533761i
$352$		0			0
$353$	4.65248			0.247626			0.123813	−	0.992306i	$-0.460488\pi$
$353$	4.65248			0.247626			0.123813	+	0.992306i	$0.460488\pi$
$354$		0			0
$355$	−15.4164			−0.818218
$356$		0			0
$357$	−12.7082			−0.672589
$358$		0			0
$359$			25.3050i			1.33554i	0.744366	+	0.667772i	$0.232752\pi$
$359$			25.3050i			1.33554i	−0.744366	+	0.667772i	$0.767248\pi$
$360$		0			0
$361$	−40.4164			−2.12718
$362$		0			0
$363$		−	6.00000i		−	0.314918i
$364$		0			0
$365$		−	24.9443i		−	1.30564i
$366$		0			0
$367$			18.0000i			0.939592i	0.882775	+	0.469796i	$0.155673\pi$
$367$			18.0000i			0.939592i	−0.882775	+	0.469796i	$0.844327\pi$
$368$		0			0
$369$			4.47214i			0.232810i
$370$		0			0
$371$	28.6525			1.48756
$372$		0			0
$373$	−3.88854			−0.201341			−0.100671	−	0.994920i	$-0.532099\pi$
$373$	−3.88854			−0.201341			−0.100671	+	0.994920i	$0.532099\pi$
$374$		0			0
$375$		−	1.52786i		−	0.0788986i
$376$		0			0
$377$	4.47214	−	3.00000i	0.230327	−	0.154508i
$378$		0			0
$379$		−	14.2918i		−	0.734120i	−0.930197	−	0.367060i	$-0.880364\pi$
$379$		−	14.2918i		−	0.734120i	0.930197	−	0.367060i	$-0.119636\pi$
$380$		0			0
$381$	12.4721			0.638967
$382$		0			0
$383$	5.34752			0.273246			0.136623	−	0.990623i	$-0.456375\pi$
$383$	5.34752			0.273246			0.136623	+	0.990623i	$0.456375\pi$
$384$		0			0
$385$		−	30.6525i		−	1.56219i
$386$		0			0
$387$		−	0.472136i		−	0.0240000i
$388$		0			0
$389$		−	13.4721i		−	0.683064i	−0.939870	−	0.341532i	$-0.889054\pi$
$389$		−	13.4721i		−	0.683064i	0.939870	−	0.341532i	$-0.110946\pi$
$390$		0			0
$391$			3.70820i			0.187532i
$392$		0			0
$393$	−12.2361			−0.617228
$394$		0			0
$395$			7.41641i			0.373160i
$396$		0			0
$397$	6.94427			0.348523			0.174262	−	0.984699i	$-0.444246\pi$
$397$	6.94427			0.348523			0.174262	+	0.984699i	$0.444246\pi$
$398$		0			0
$399$	−32.6525			−1.63467
$400$		0			0
$401$	33.7082			1.68331			0.841654	−	0.540018i	$-0.181582\pi$
$401$	33.7082			1.68331			0.841654	+	0.540018i	$0.181582\pi$
$402$		0			0
$403$		−	7.23607i		−	0.360454i
$404$		0			0
$405$	3.23607			0.160802
$406$		0			0
$407$	10.6525			0.528024
$408$		0			0
$409$		−	10.5836i		−	0.523325i	−0.965159	−	0.261662i	$-0.915729\pi$
$409$		−	10.5836i		−	0.523325i	0.965159	−	0.261662i	$-0.0842707\pi$
$410$		0			0
$411$	−3.52786			−0.174017
$412$		0			0
$413$	−37.8885			−1.86437
$414$		0			0
$415$	19.4164			0.953114
$416$		0			0
$417$		−	6.70820i		−	0.328502i
$418$		0			0
$419$	−3.81966			−0.186603			−0.0933013	−	0.995638i	$-0.529742\pi$
$419$	−3.81966			−0.186603			−0.0933013	+	0.995638i	$0.529742\pi$
$420$		0			0
$421$			20.0000i			0.974740i	0.873195	+	0.487370i	$0.162044\pi$
$421$			20.0000i			0.974740i	−0.873195	+	0.487370i	$0.837956\pi$
$422$		0			0
$423$		−	10.2361i		−	0.497695i
$424$		0			0
$425$		−	16.4164i		−	0.796313i
$426$		0			0
$427$			11.7082i			0.566600i
$428$		0			0
$429$	2.23607			0.107958
$430$		0			0
$431$	−14.3607			−0.691730			−0.345865	−	0.938284i	$-0.612414\pi$
$431$	−14.3607			−0.691730			−0.345865	+	0.938284i	$0.612414\pi$
$432$		0			0
$433$		−	14.2918i		−	0.686820i	−0.939186	−	0.343410i	$-0.888418\pi$
$433$		−	14.2918i		−	0.686820i	0.939186	−	0.343410i	$-0.111582\pi$
$434$		0			0
$435$	9.70820	+	14.4721i	0.465473	+	0.693886i
$436$		0			0
$437$			9.52786i			0.455780i
$438$		0			0
$439$	−13.2918			−0.634383			−0.317191	−	0.948362i	$-0.602740\pi$
$439$	−13.2918			−0.634383			−0.317191	+	0.948362i	$0.602740\pi$
$440$		0			0
$441$	−10.9443			−0.521156
$442$		0			0
$443$		−	9.65248i		−	0.458603i	−0.973355	−	0.229301i	$-0.926356\pi$
$443$		−	9.65248i		−	0.458603i	0.973355	−	0.229301i	$-0.0736442\pi$
$444$		0			0
$445$		−	0.180340i		−	0.00854893i
$446$		0			0
$447$			5.52786i			0.261459i
$448$		0			0
$449$			18.8885i			0.891405i	0.895181	+	0.445703i	$0.147046\pi$
$449$			18.8885i			0.891405i	−0.895181	+	0.445703i	$0.852954\pi$
$450$		0			0
$451$	−10.0000			−0.470882
$452$		0			0
$453$			12.0000i			0.563809i
$454$		0			0
$455$	−13.7082			−0.642651
$456$		0			0
$457$	6.41641			0.300147			0.150073	−	0.988675i	$-0.452049\pi$
$457$	6.41641			0.300147			0.150073	+	0.988675i	$0.452049\pi$
$458$		0			0
$459$	3.00000			0.140028
$460$		0			0
$461$		−	18.9443i		−	0.882323i	−0.897428	−	0.441161i	$-0.854567\pi$
$461$		−	18.9443i		−	0.882323i	0.897428	−	0.441161i	$-0.145433\pi$
$462$		0			0
$463$	−0.708204			−0.0329130			−0.0164565	−	0.999865i	$-0.505239\pi$
$463$	−0.708204			−0.0329130			−0.0164565	+	0.999865i	$0.505239\pi$
$464$		0			0
$465$	23.4164			1.08591
$466$		0			0
$467$			13.5279i			0.625995i	0.949754	+	0.312997i	$0.101333\pi$
$467$			13.5279i			0.625995i	−0.949754	+	0.312997i	$0.898667\pi$
$468$		0			0
$469$	−19.9443			−0.920941
$470$		0			0
$471$	2.00000			0.0921551
$472$		0			0
$473$	1.05573			0.0485424
$474$		0			0
$475$		−	42.1803i		−	1.93537i
$476$		0			0
$477$	−6.76393			−0.309699
$478$		0			0
$479$		−	10.4721i		−	0.478484i	−0.970960	−	0.239242i	$-0.923101\pi$
$479$		−	10.4721i		−	0.478484i	0.970960	−	0.239242i	$-0.0768989\pi$
$480$		0			0
$481$		−	4.76393i		−	0.217217i
$482$		0			0
$483$			5.23607i			0.238249i
$484$		0			0
$485$			58.8328i			2.67146i
$486$		0			0
$487$	−20.3607			−0.922630			−0.461315	−	0.887236i	$-0.652622\pi$
$487$	−20.3607			−0.922630			−0.461315	+	0.887236i	$0.652622\pi$
$488$		0			0
$489$	−7.41641			−0.335382
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	9.00000	+	13.4164i	0.405340	+	0.604245i
$494$		0			0
$495$			7.23607i			0.325237i
$496$		0			0
$497$	−20.1803			−0.905212
$498$		0			0
$499$	−21.1803			−0.948162			−0.474081	−	0.880481i	$-0.657220\pi$
$499$	−21.1803			−0.948162			−0.474081	+	0.880481i	$0.657220\pi$
$500$		0			0
$501$			15.8885i			0.709848i
$502$		0			0
$503$		−	25.1803i		−	1.12274i	−0.827566	−	0.561368i	$-0.810275\pi$
$503$		−	25.1803i		−	1.12274i	0.827566	−	0.561368i	$-0.189725\pi$
$504$		0			0
$505$		−	12.7639i		−	0.567988i
$506$		0			0
$507$			12.0000i			0.532939i
$508$		0			0
$509$	−26.8328			−1.18934			−0.594672	−	0.803969i	$-0.702718\pi$
$509$	−26.8328			−1.18934			−0.594672	+	0.803969i	$0.702718\pi$
$510$		0			0
$511$		−	32.6525i		−	1.44446i
$512$		0			0
$513$	7.70820			0.340326
$514$		0			0
$515$	62.8328			2.76874
$516$		0			0
$517$	22.8885			1.00664
$518$		0			0
$519$			6.00000i			0.263371i
$520$		0			0
$521$	−3.52786			−0.154559			−0.0772793	−	0.997009i	$-0.524623\pi$
$521$	−3.52786			−0.154559			−0.0772793	+	0.997009i	$0.524623\pi$
$522$		0			0
$523$	−33.0689			−1.44600			−0.723001	−	0.690847i	$-0.757238\pi$
$523$	−33.0689			−1.44600			−0.723001	+	0.690847i	$0.757238\pi$
$524$		0			0
$525$		−	23.1803i		−	1.01167i
$526$		0			0
$527$	21.7082			0.945624
$528$		0			0
$529$	−21.4721			−0.933571
$530$		0			0
$531$	8.94427			0.388148
$532$		0			0
$533$			4.47214i			0.193710i
$534$		0			0
$535$	−2.47214			−0.106880
$536$		0			0
$537$		−	17.2361i		−	0.743791i
$538$		0			0
$539$		−	24.4721i		−	1.05409i
$540$		0			0
$541$		−	10.6525i		−	0.457986i	−0.973428	−	0.228993i	$-0.926457\pi$
$541$		−	10.6525i		−	0.457986i	0.973428	−	0.228993i	$-0.0735432\pi$
$542$		0			0
$543$		−	20.4164i		−	0.876152i
$544$		0			0
$545$	16.1803			0.693090
$546$		0			0
$547$	6.34752			0.271401			0.135700	−	0.990750i	$-0.456672\pi$
$547$	6.34752			0.271401			0.135700	+	0.990750i	$0.456672\pi$
$548$		0			0
$549$		−	2.76393i		−	0.117962i
$550$		0			0
$551$	23.1246	+	34.4721i	0.985142	+	1.46856i
$552$		0			0
$553$			9.70820i			0.412835i
$554$		0			0
$555$	15.4164			0.654390
$556$		0			0
$557$	−16.4721			−0.697947			−0.348973	−	0.937133i	$-0.613470\pi$
$557$	−16.4721			−0.697947			−0.348973	+	0.937133i	$0.613470\pi$
$558$		0			0
$559$		−	0.472136i		−	0.0199692i
$560$		0			0
$561$			6.70820i			0.283221i
$562$		0			0
$563$		−	13.0689i		−	0.550788i	−0.961331	−	0.275394i	$-0.911192\pi$
$563$		−	13.0689i		−	0.550788i	0.961331	−	0.275394i	$-0.0888083\pi$
$564$		0			0
$565$			14.6525i			0.616434i
$566$		0			0
$567$	4.23607			0.177898
$568$		0			0
$569$			21.0000i			0.880366i	0.897908	+	0.440183i	$0.145086\pi$
$569$			21.0000i			0.880366i	−0.897908	+	0.440183i	$0.854914\pi$
$570$		0			0
$571$	−18.8328			−0.788129			−0.394064	−	0.919083i	$-0.628931\pi$
$571$	−18.8328			−0.788129			−0.394064	+	0.919083i	$0.628931\pi$
$572$		0			0
$573$	−8.94427			−0.373652
$574$		0			0
$575$	−6.76393			−0.282075
$576$		0			0
$577$		−	24.8328i		−	1.03380i	−0.856045	−	0.516902i	$-0.827085\pi$
$577$		−	24.8328i		−	1.03380i	0.856045	−	0.516902i	$-0.172915\pi$
$578$		0			0
$579$	17.4164			0.723801
$580$		0			0
$581$	25.4164			1.05445
$582$		0			0
$583$		−	15.1246i		−	0.626397i
$584$		0			0
$585$	3.23607			0.133795
$586$		0			0
$587$	0.944272			0.0389743			0.0194871	−	0.999810i	$-0.493797\pi$
$587$	0.944272			0.0389743			0.0194871	+	0.999810i	$0.493797\pi$
$588$		0			0
$589$	55.7771			2.29825
$590$		0			0
$591$			3.70820i			0.152535i
$592$		0			0
$593$	40.1803			1.65001			0.825004	−	0.565126i	$-0.191173\pi$
$593$	40.1803			1.65001			0.825004	+	0.565126i	$0.191173\pi$
$594$		0			0
$595$		−	41.1246i		−	1.68594i
$596$		0			0
$597$		−	20.1246i		−	0.823646i
$598$		0			0
$599$			1.76393i			0.0720723i	0.999350	+	0.0360362i	$0.0114731\pi$
$599$			1.76393i			0.0720723i	−0.999350	+	0.0360362i	$0.988527\pi$
$600$		0			0
$601$		−	18.5410i		−	0.756304i	−0.925744	−	0.378152i	$-0.876560\pi$
$601$		−	18.5410i		−	0.756304i	0.925744	−	0.378152i	$-0.123440\pi$
$602$		0			0
$603$	4.70820			0.191733
$604$		0			0
$605$	19.4164			0.789389
$606$		0			0
$607$		−	25.4164i		−	1.03162i	−0.856703	−	0.515810i	$-0.827491\pi$
$607$		−	25.4164i		−	1.03162i	0.856703	−	0.515810i	$-0.172509\pi$
$608$		0			0
$609$	12.7082	+	18.9443i	0.514962	+	0.767661i
$610$		0			0
$611$		−	10.2361i		−	0.414107i
$612$		0			0
$613$	−24.4164			−0.986169			−0.493085	−	0.869981i	$-0.664131\pi$
$613$	−24.4164			−0.986169			−0.493085	+	0.869981i	$0.664131\pi$
$614$		0			0
$615$	−14.4721			−0.583573
$616$		0			0
$617$			19.8885i			0.800683i	0.916366	+	0.400341i	$0.131108\pi$
$617$			19.8885i			0.800683i	−0.916366	+	0.400341i	$0.868892\pi$
$618$		0			0
$619$			32.5410i			1.30793i	0.756523	+	0.653967i	$0.226896\pi$
$619$			32.5410i			1.30793i	−0.756523	+	0.653967i	$0.773104\pi$
$620$		0			0
$621$		−	1.23607i		−	0.0496017i
$622$		0			0
$623$		−	0.236068i		−	0.00945786i
$624$		0			0
$625$	−22.4164			−0.896656
$626$		0			0
$627$			17.2361i			0.688342i
$628$		0			0
$629$	14.2918			0.569851
$630$		0			0
$631$	−28.7082			−1.14286			−0.571428	−	0.820652i	$-0.693610\pi$
$631$	−28.7082			−1.14286			−0.571428	+	0.820652i	$0.693610\pi$
$632$		0			0
$633$	−10.0000			−0.397464
$634$		0			0
$635$			40.3607i			1.60166i
$636$		0			0
$637$	−10.9443			−0.433628
$638$		0			0
$639$	4.76393			0.188458
$640$		0			0
$641$		−	19.4721i		−	0.769103i	−0.923104	−	0.384552i	$-0.874356\pi$
$641$		−	19.4721i		−	0.769103i	0.923104	−	0.384552i	$-0.125644\pi$
$642$		0			0
$643$	−0.708204			−0.0279288			−0.0139644	−	0.999902i	$-0.504445\pi$
$643$	−0.708204			−0.0279288			−0.0139644	+	0.999902i	$0.504445\pi$
$644$		0			0
$645$	1.52786			0.0601596
$646$		0			0
$647$	−17.5967			−0.691800			−0.345900	−	0.938271i	$-0.612426\pi$
$647$	−17.5967			−0.691800			−0.345900	+	0.938271i	$0.612426\pi$
$648$		0			0
$649$			20.0000i			0.785069i
$650$		0			0
$651$	30.6525			1.20137
$652$		0			0
$653$		−	32.3050i		−	1.26419i	−0.774891	−	0.632095i	$-0.782195\pi$
$653$		−	32.3050i		−	1.26419i	0.774891	−	0.632095i	$-0.217805\pi$
$654$		0			0
$655$		−	39.5967i		−	1.54717i
$656$		0			0
$657$			7.70820i			0.300726i
$658$		0			0
$659$			26.2361i			1.02201i	0.859577	+	0.511006i	$0.170727\pi$
$659$			26.2361i			1.02201i	−0.859577	+	0.511006i	$0.829273\pi$
$660$		0			0
$661$	−29.8328			−1.16036			−0.580181	−	0.814488i	$-0.697018\pi$
$661$	−29.8328			−1.16036			−0.580181	+	0.814488i	$0.697018\pi$
$662$		0			0
$663$	3.00000			0.116510
$664$		0			0
$665$		−	105.666i		−	4.09754i
$666$		0			0
$667$	5.52786	−	3.70820i	0.214040	−	0.143582i
$668$		0			0
$669$		−	12.7082i		−	0.491328i
$670$		0			0
$671$	6.18034			0.238589
$672$		0			0
$673$	−4.41641			−0.170240			−0.0851200	−	0.996371i	$-0.527127\pi$
$673$	−4.41641			−0.170240			−0.0851200	+	0.996371i	$0.527127\pi$
$674$		0			0
$675$			5.47214i			0.210623i
$676$		0			0
$677$			18.0557i			0.693938i	0.937877	+	0.346969i	$0.112789\pi$
$677$			18.0557i			0.693938i	−0.937877	+	0.346969i	$0.887211\pi$
$678$		0			0
$679$			77.0132i			2.95549i
$680$		0			0
$681$			9.05573i			0.347016i
$682$		0			0
$683$	−32.9443			−1.26058			−0.630289	−	0.776361i	$-0.717064\pi$
$683$	−32.9443			−1.26058			−0.630289	+	0.776361i	$0.717064\pi$
$684$		0			0
$685$		−	11.4164i		−	0.436199i
$686$		0			0
$687$	22.1803			0.846233
$688$		0			0
$689$	−6.76393			−0.257685
$690$		0			0
$691$	30.2361			1.15023			0.575117	−	0.818071i	$-0.304956\pi$
$691$	30.2361			1.15023			0.575117	+	0.818071i	$0.304956\pi$
$692$		0			0
$693$			9.47214i			0.359817i
$694$		0			0
$695$	21.7082			0.823439
$696$		0			0
$697$	−13.4164			−0.508183
$698$		0			0
$699$			10.4721i			0.396093i
$700$		0			0
$701$	8.18034			0.308967			0.154484	−	0.987995i	$-0.450629\pi$
$701$	8.18034			0.308967			0.154484	+	0.987995i	$0.450629\pi$
$702$		0			0
$703$	36.7214			1.38497
$704$		0			0
$705$	33.1246			1.24755
$706$		0			0
$707$		−	16.7082i		−	0.628377i
$708$		0			0
$709$	−13.4164			−0.503864			−0.251932	−	0.967745i	$-0.581066\pi$
$709$	−13.4164			−0.503864			−0.251932	+	0.967745i	$0.581066\pi$
$710$		0			0
$711$		−	2.29180i		−	0.0859491i
$712$		0			0
$713$		−	8.94427i		−	0.334966i
$714$		0			0
$715$			7.23607i			0.270614i
$716$		0			0
$717$			26.1803i			0.977723i
$718$		0			0
$719$	30.0000			1.11881			0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000			1.11881			0.559406	+	0.828894i	$0.311029\pi$
$720$		0			0
$721$	82.2492			3.06312
$722$		0			0
$723$			3.00000i			0.111571i
$724$		0			0
$725$	−24.4721	+	16.4164i	−0.908872	+	0.609690i
$726$		0			0
$727$		−	5.81966i		−	0.215839i	−0.994160	−	0.107920i	$-0.965581\pi$
$727$		−	5.81966i		−	0.215839i	0.994160	−	0.107920i	$-0.0344189\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	1.41641			0.0523877
$732$		0			0
$733$			8.47214i			0.312925i	0.987684	+	0.156463i	$0.0500091\pi$
$733$			8.47214i			0.312925i	−0.987684	+	0.156463i	$0.949991\pi$
$734$		0			0
$735$		−	35.4164i		−	1.30635i
$736$		0			0
$737$			10.5279i			0.387799i
$738$		0			0
$739$		−	12.1803i		−	0.448061i	−0.974582	−	0.224031i	$-0.928078\pi$
$739$		−	12.1803i		−	0.448061i	0.974582	−	0.224031i	$-0.0719215\pi$
$740$		0			0
$741$	7.70820			0.283168
$742$		0			0
$743$		−	43.0689i		−	1.58004i	−0.613078	−	0.790022i	$-0.710069\pi$
$743$		−	43.0689i		−	1.58004i	0.613078	−	0.790022i	$-0.289931\pi$
$744$		0			0
$745$	−17.8885			−0.655386
$746$		0			0
$747$	−6.00000			−0.219529
$748$		0			0
$749$	−3.23607			−0.118243
$750$		0			0
$751$		−	26.8328i		−	0.979143i	−0.871963	−	0.489572i	$-0.837153\pi$
$751$		−	26.8328i		−	0.979143i	0.871963	−	0.489572i	$-0.162847\pi$
$752$		0			0
$753$	12.2361			0.445907
$754$		0			0
$755$	−38.8328			−1.41327
$756$		0			0
$757$		−	38.0000i		−	1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$		−	38.0000i		−	1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$		0			0
$759$	2.76393			0.100324
$760$		0			0
$761$	2.65248			0.0961522			0.0480761	−	0.998844i	$-0.484691\pi$
$761$	2.65248			0.0961522			0.0480761	+	0.998844i	$0.484691\pi$
$762$		0			0
$763$	21.1803			0.766780
$764$		0			0
$765$			9.70820i			0.351001i
$766$		0			0
$767$	8.94427			0.322959
$768$		0			0
$769$		−	40.1803i		−	1.44894i	−0.689306	−	0.724470i	$-0.742085\pi$
$769$		−	40.1803i		−	1.44894i	0.689306	−	0.724470i	$-0.257915\pi$
$770$		0			0
$771$			26.4721i			0.953371i
$772$		0			0
$773$			31.8885i			1.14695i	0.819223	+	0.573476i	$0.194405\pi$
$773$			31.8885i			1.14695i	−0.819223	+	0.573476i	$0.805595\pi$
$774$		0			0
$775$			39.5967i			1.42236i
$776$		0			0
$777$	20.1803			0.723966
$778$		0			0
$779$	−34.4721			−1.23509
$780$		0			0
$781$			10.6525i			0.381176i
$782$		0			0
$783$	−3.00000	−	4.47214i	−0.107211	−	0.159821i
$784$		0			0
$785$			6.47214i			0.231000i
$786$		0			0
$787$	12.0000			0.427754			0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000			0.427754			0.213877	+	0.976861i	$0.431391\pi$
$788$		0			0
$789$	4.94427			0.176021
$790$		0			0
$791$			19.1803i			0.681974i
$792$		0			0
$793$		−	2.76393i		−	0.0981501i
$794$		0			0
$795$		−	21.8885i		−	0.776307i
$796$		0			0
$797$		−	44.8328i		−	1.58806i	−0.607879	−	0.794030i	$-0.707979\pi$
$797$		−	44.8328i		−	1.58806i	0.607879	−	0.794030i	$-0.292021\pi$
$798$		0			0
$799$	30.7082			1.08638
$800$		0			0
$801$			0.0557281i			0.00196906i
$802$		0			0
$803$	−17.2361			−0.608248
$804$		0			0
$805$	−16.9443			−0.597207
$806$		0			0
$807$	−21.3607			−0.751932
$808$		0			0
$809$			43.3607i			1.52448i	0.647294	+	0.762240i	$0.275900\pi$
$809$			43.3607i			1.52448i	−0.647294	+	0.762240i	$0.724100\pi$
$810$		0			0
$811$	−46.5967			−1.63623			−0.818117	−	0.575052i	$-0.804982\pi$
$811$	−46.5967			−1.63623			−0.818117	+	0.575052i	$0.804982\pi$
$812$		0			0
$813$	23.4164			0.821249
$814$		0			0
$815$		−	24.0000i		−	0.840683i
$816$		0			0
$817$	3.63932			0.127324
$818$		0			0
$819$	4.23607			0.148020
$820$		0			0
$821$	−12.4721			−0.435281			−0.217640	−	0.976029i	$-0.569836\pi$
$821$	−12.4721			−0.435281			−0.217640	+	0.976029i	$0.569836\pi$
$822$		0			0
$823$		−	26.3607i		−	0.918876i	−0.888210	−	0.459438i	$-0.848051\pi$
$823$		−	26.3607i		−	0.918876i	0.888210	−	0.459438i	$-0.151949\pi$
$824$		0			0
$825$	−12.2361			−0.426005
$826$		0			0
$827$		−	29.8885i		−	1.03933i	−0.854371	−	0.519663i	$-0.826057\pi$
$827$		−	29.8885i		−	1.03933i	0.854371	−	0.519663i	$-0.173943\pi$
$828$		0			0
$829$			27.3050i			0.948340i	0.880433	+	0.474170i	$0.157252\pi$
$829$			27.3050i			0.948340i	−0.880433	+	0.474170i	$0.842748\pi$
$830$		0			0
$831$		−	6.41641i		−	0.222583i
$832$		0			0
$833$		−	32.8328i		−	1.13759i
$834$		0			0
$835$	−51.4164			−1.77934
$836$		0			0
$837$	−7.23607			−0.250115
$838$		0			0
$839$			0.708204i			0.0244499i	0.999925	+	0.0122250i	$0.00389142\pi$
$839$			0.708204i			0.0244499i	−0.999925	+	0.0122250i	$0.996109\pi$
$840$		0			0
$841$	11.0000	−	26.8328i	0.379310	−	0.925270i
$842$		0			0
$843$			1.41641i			0.0487837i
$844$		0			0
$845$	−38.8328			−1.33589
$846$		0			0
$847$	25.4164			0.873318
$848$		0			0
$849$		−	13.8885i		−	0.476654i
$850$		0			0
$851$		−	5.88854i		−	0.201857i
$852$		0			0
$853$		−	45.1935i		−	1.54740i	−0.633555	−	0.773698i	$-0.718405\pi$
$853$		−	45.1935i		−	1.54740i	0.633555	−	0.773698i	$-0.281595\pi$
$854$		0			0
$855$			24.9443i			0.853076i
$856$		0			0
$857$	37.5967			1.28428			0.642140	−	0.766587i	$-0.278047\pi$
$857$	37.5967			1.28428			0.642140	+	0.766587i	$0.278047\pi$
$858$		0			0
$859$			36.3607i			1.24061i	0.784361	+	0.620305i	$0.212991\pi$
$859$			36.3607i			1.24061i	−0.784361	+	0.620305i	$0.787009\pi$
$860$		0			0
$861$	−18.9443			−0.645619
$862$		0			0
$863$	46.6525			1.58807			0.794034	−	0.607873i	$-0.207977\pi$
$863$	46.6525			1.58807			0.794034	+	0.607873i	$0.207977\pi$
$864$		0			0
$865$	−19.4164			−0.660178
$866$		0			0
$867$		−	8.00000i		−	0.271694i
$868$		0			0
$869$	5.12461			0.173841
$870$		0			0
$871$	4.70820			0.159531
$872$		0			0
$873$		−	18.1803i		−	0.615311i
$874$		0			0
$875$	6.47214			0.218798
$876$		0			0
$877$	32.4721			1.09651			0.548253	−	0.836312i	$-0.315293\pi$
$877$	32.4721			1.09651			0.548253	+	0.836312i	$0.315293\pi$
$878$		0			0
$879$	−29.9443			−1.01000
$880$		0			0
$881$		−	55.2492i		−	1.86139i	−0.365792	−	0.930697i	$-0.619202\pi$
$881$		−	55.2492i		−	1.86139i	0.365792	−	0.930697i	$-0.380798\pi$
$882$		0			0
$883$	−34.2492			−1.15258			−0.576289	−	0.817246i	$-0.695500\pi$
$883$	−34.2492			−1.15258			−0.576289	+	0.817246i	$0.695500\pi$
$884$		0			0
$885$			28.9443i			0.972951i
$886$		0			0
$887$			17.0689i			0.573117i	0.958063	+	0.286559i	$0.0925113\pi$
$887$			17.0689i			0.573117i	−0.958063	+	0.286559i	$0.907489\pi$
$888$		0			0
$889$			52.8328i			1.77196i
$890$		0			0
$891$		−	2.23607i		−	0.0749111i
$892$		0			0
$893$	78.9017			2.64034
$894$		0			0
$895$	55.7771			1.86442
$896$		0			0
$897$		−	1.23607i		−	0.0412711i
$898$		0			0
$899$	−21.7082	−	32.3607i	−0.724009	−	1.07929i
$900$		0			0
$901$		−	20.2918i		−	0.676018i
$902$		0			0
$903$	2.00000			0.0665558
$904$		0			0
$905$	66.0689			2.19620
$906$		0			0
$907$		−	26.4721i		−	0.878993i	−0.898244	−	0.439496i	$-0.855157\pi$
$907$		−	26.4721i		−	0.878993i	0.898244	−	0.439496i	$-0.144843\pi$
$908$		0			0
$909$			3.94427i			0.130823i
$910$		0			0
$911$			51.1803i			1.69568i	0.530252	+	0.847840i	$0.322097\pi$
$911$			51.1803i			1.69568i	−0.530252	+	0.847840i	$0.677903\pi$
$912$		0			0
$913$		−	13.4164i		−	0.444018i
$914$		0			0
$915$	8.94427			0.295689
$916$		0			0
$917$		−	51.8328i		−	1.71167i
$918$		0			0
$919$	25.6525			0.846197			0.423099	−	0.906084i	$-0.360942\pi$
$919$	25.6525			0.846197			0.423099	+	0.906084i	$0.360942\pi$
$920$		0			0
$921$	−28.1803			−0.928574
$922$		0			0
$923$	4.76393			0.156807
$924$		0			0
$925$			26.0689i			0.857140i
$926$		0			0
$927$	−19.4164			−0.637719
$928$		0			0
$929$	−32.3607			−1.06172			−0.530860	−	0.847460i	$-0.678131\pi$
$929$	−32.3607			−1.06172			−0.530860	+	0.847460i	$0.678131\pi$
$930$		0			0
$931$		−	84.3607i		−	2.76481i
$932$		0			0
$933$	15.6525			0.512439
$934$		0			0
$935$	−21.7082			−0.709934
$936$		0			0
$937$	33.0000			1.07806			0.539032	−	0.842286i	$-0.318790\pi$
$937$	33.0000			1.07806			0.539032	+	0.842286i	$0.318790\pi$
$938$		0			0
$939$			5.47214i			0.178576i
$940$		0			0
$941$	−44.8328			−1.46151			−0.730754	−	0.682641i	$-0.760831\pi$
$941$	−44.8328			−1.46151			−0.730754	+	0.682641i	$0.760831\pi$
$942$		0			0
$943$			5.52786i			0.180012i
$944$		0			0
$945$			13.7082i			0.445928i
$946$		0			0
$947$			16.8197i			0.546566i	0.961934	+	0.273283i	$0.0881095\pi$
$947$			16.8197i			0.546566i	−0.961934	+	0.273283i	$0.911891\pi$
$948$		0			0
$949$			7.70820i			0.250219i
$950$		0			0
$951$	23.8328			0.772832
$952$		0			0
$953$	−59.6656			−1.93276			−0.966380	−	0.257119i	$-0.917227\pi$
$953$	−59.6656			−1.93276			−0.966380	+	0.257119i	$0.917227\pi$
$954$		0			0
$955$		−	28.9443i		−	0.936615i
$956$		0			0
$957$	10.0000	−	6.70820i	0.323254	−	0.216845i
$958$		0			0
$959$		−	14.9443i		−	0.482576i
$960$		0			0
$961$	−21.3607			−0.689054
$962$		0			0
$963$	0.763932			0.0246174
$964$		0			0
$965$			56.3607i			1.81431i
$966$		0			0
$967$		−	27.1246i		−	0.872269i	−0.899882	−	0.436134i	$-0.856347\pi$
$967$		−	27.1246i		−	0.872269i	0.899882	−	0.436134i	$-0.143653\pi$
$968$		0			0
$969$			23.1246i			0.742870i
$970$		0			0
$971$		−	16.5836i		−	0.532193i	−0.963946	−	0.266096i	$-0.914266\pi$
$971$		−	16.5836i		−	0.532193i	0.963946	−	0.266096i	$-0.0857340\pi$
$972$		0			0
$973$	28.4164			0.910988
$974$		0			0
$975$			5.47214i			0.175249i
$976$		0			0
$977$	51.0132			1.63206			0.816028	−	0.578013i	$-0.196172\pi$
$977$	51.0132			1.63206			0.816028	+	0.578013i	$0.196172\pi$
$978$		0			0
$979$	−0.124612			−0.00398261
$980$		0			0
$981$	−5.00000			−0.159638
$982$		0			0
$983$			13.8885i			0.442976i	0.975163	+	0.221488i	$0.0710913\pi$
$983$			13.8885i			0.442976i	−0.975163	+	0.221488i	$0.928909\pi$
$984$		0			0
$985$	−12.0000			−0.382352
$986$		0			0
$987$	43.3607			1.38019
$988$		0			0
$989$		−	0.583592i		−	0.0185572i
$990$		0			0
$991$	21.2918			0.676356			0.338178	−	0.941082i	$-0.390189\pi$
$991$	21.2918			0.676356			0.338178	+	0.941082i	$0.390189\pi$
$992$		0			0
$993$	8.29180			0.263132
$994$		0			0
$995$	65.1246			2.06459
$996$		0			0
$997$		−	24.5836i		−	0.778570i	−0.921117	−	0.389285i	$-0.872722\pi$
$997$		−	24.5836i		−	0.778570i	0.921117	−	0.389285i	$-0.127278\pi$
$998$		0			0
$999$	−4.76393			−0.150724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1392.2.o.i.289.2		4
3.2	odd	2		4176.2.o.l.289.2		4
4.3	odd	2		87.2.c.a.28.1	✓	4
12.11	even	2		261.2.c.b.28.4		4
20.3	even	4		2175.2.f.a.724.2		4
20.7	even	4		2175.2.f.b.724.3		4
20.19	odd	2		2175.2.d.e.376.4		4
29.28	even	2	inner	1392.2.o.i.289.4		4
87.86	odd	2		4176.2.o.l.289.1		4
116.75	even	4		2523.2.a.d.1.1		2
116.99	even	4		2523.2.a.e.1.2		2
116.115	odd	2		87.2.c.a.28.4	yes	4
348.191	odd	4		7569.2.a.n.1.2		2
348.215	odd	4		7569.2.a.f.1.1		2
348.347	even	2		261.2.c.b.28.1		4
580.347	even	4		2175.2.f.a.724.1		4
580.463	even	4		2175.2.f.b.724.4		4
580.579	odd	2		2175.2.d.e.376.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.c.a.28.1	✓	4	4.3	odd	2
87.2.c.a.28.4	yes	4	116.115	odd	2
261.2.c.b.28.1		4	348.347	even	2
261.2.c.b.28.4		4	12.11	even	2
1392.2.o.i.289.2		4	1.1	even	1	trivial
1392.2.o.i.289.4		4	29.28	even	2	inner
2175.2.d.e.376.1		4	580.579	odd	2
2175.2.d.e.376.4		4	20.19	odd	2
2175.2.f.a.724.1		4	580.347	even	4
2175.2.f.a.724.2		4	20.3	even	4
2175.2.f.b.724.3		4	20.7	even	4
2175.2.f.b.724.4		4	580.463	even	4
2523.2.a.d.1.1		2	116.75	even	4
2523.2.a.e.1.2		2	116.99	even	4
4176.2.o.l.289.1		4	87.86	odd	2
4176.2.o.l.289.2		4	3.2	odd	2
7569.2.a.f.1.1		2	348.215	odd	4
7569.2.a.n.1.2		2	348.191	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1392 = 2^{4} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1392.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.23607 q^{5} +4.23607 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.23607 q^{5} +4.23607 q^{7} -1.00000 q^{9} -2.23607i q^{11} -1.00000 q^{13} -3.23607i q^{15} -3.00000i q^{17} -7.70820i q^{19} -4.23607i q^{21} -1.23607 q^{23} +5.47214 q^{25} +1.00000i q^{27} +(-4.47214 + 3.00000i) q^{29} +7.23607i q^{31} -2.23607 q^{33} +13.7082 q^{35} +4.76393i q^{37} +1.00000i q^{39} -4.47214i q^{41} +0.472136i q^{43} -3.23607 q^{45} +10.2361i q^{47} +10.9443 q^{49} -3.00000 q^{51} +6.76393 q^{53} -7.23607i q^{55} -7.70820 q^{57} -8.94427 q^{59} +2.76393i q^{61} -4.23607 q^{63} -3.23607 q^{65} -4.70820 q^{67} +1.23607i q^{69} -4.76393 q^{71} -7.70820i q^{73} -5.47214i q^{75} -9.47214i q^{77} +2.29180i q^{79} +1.00000 q^{81} +6.00000 q^{83} -9.70820i q^{85} +(3.00000 + 4.47214i) q^{87} -0.0557281i q^{89} -4.23607 q^{91} +7.23607 q^{93} -24.9443i q^{95} +18.1803i q^{97} +2.23607i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 8 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 8 * q^7 - 4 * q^9	\( 4 q + 4 q^{5} + 8 q^{7} - 4 q^{9} - 4 q^{13} + 4 q^{23} + 4 q^{25} + 28 q^{35} - 4 q^{45} + 8 q^{49} - 12 q^{51} + 36 q^{53} - 4 q^{57} - 8 q^{63} - 4 q^{65} + 8 q^{67} - 28 q^{71} + 4 q^{81} + 24 q^{83} + 12 q^{87} - 8 q^{91} + 20 q^{93}+O(q^{100}) \) 4 * q + 4 * q^5 + 8 * q^7 - 4 * q^9 - 4 * q^13 + 4 * q^23 + 4 * q^25 + 28 * q^35 - 4 * q^45 + 8 * q^49 - 12 * q^51 + 36 * q^53 - 4 * q^57 - 8 * q^63 - 4 * q^65 + 8 * q^67 - 28 * q^71 + 4 * q^81 + 24 * q^83 + 12 * q^87 - 8 * q^91 + 20 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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